**European option pricing with transaction costs and stochastic**

**volatility: an asymptotic analysis**

R. E. Caflisch

*Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, USA*
rcaflisch@ipam.ucla.edu

G. Gambino and M. Sammartino

*Department of Mathematics, University of Palermo, Palermo, Italy*
gaetana@math.unipa.it marco@math.unipa.it

and C. Sgarra∗

*Department of Mathematics, Politecnico di Milano, Milan, Italy*
∗_{Corresponding author: carlo.sgarra@polimi.it}

[Received on 3 April 2013; revised on 12 November 2013; accepted on 2 June 2014]

**1. Introduction**

The intrinsic limitations of the Black–Scholes model in describing real markets behaviour are very well known. Among the main assumptions underlying this model, the most relevant ones are probably constant volatility and no transaction costs. In this paper, we are going to consider the pricing problem of a European option in a model in which both proportional transaction costs are taken into account and the volatility is assumed to evolve according to a stochastic process of the Ornstein–Uhlenbeck type. To analyse this situation we shall follow a utility maximization procedure, following the seminal paper of Davis et al. (1993).

If one uses the following utility function*U:*

*U(x) = 1 − exp (−γ x),*

where* γ expresses the risk aversion of the investor, one gets, as the result of this analysis, a non-linear *
partial differential equation (PDE) for the expected value of the utility-maximized wealth held in the
underlying asset of the option.

The pricing of a European option in the presence of small transaction costs was considered in Whalley & Wilmott (1997), where a correction term to the Black and Scholes pricing formula was

derived.* This correction term was found to be O(ε*2*/3), where ε* denotes the asymptotic expansion
param-eter. Moreover in Whalley & Wilmott (1997) it was found that the optimal hedging strategy
consists in not transacting when the process driving the stock price is in a strip around the classical
Black and Scholes delta—hedging formula, and in rebalancing the portfolio (selling or buying stocks)
to keep the process inside the strip of no transaction. The width of the no transaction strip was found to
be* O(ε*2*/3 _{). In Whalley & Wilmott (1997), the volatility has been supposed to be constant.}*

A wide literature is available on the optimal consumption-investment problem for models with trans-action costs: we just mention two pioneer contributions on the subject, the paper by Davis & Norman (1990), and the paper by Cvitanic & Karatzas (1996). More recently, ksendal & Sulem (2002) inves-tigated the optimal consumption problem in a model including both fixed and proportional transac-tion costs, while Muthuraman & Kumar (2006) studied the optimal investment problem in a multi-dimensional setting. The problem of hedging contingent claims for models with transaction costs has been investigated in several papers: Albanese & Tompaidis (2008), Clewlow & Hodges (1997), Kallsen & Muhle-Karbe (2013) and Zakamouline (2006a,b).

The pricing of a European option with fast mean-reverting stochastic volatility was considered in a series of papers (see e.g. Fouque et al., 2000, 2003, 2001, 2011; Fouque & Lorig, 2011). In the above-mentioned papers, the authors found the pricing formula whose leading order term is the classical Black and Scholes formula with averaged volatility. The correction term was order the square root of the characteristic time scale of the process driving the volatility. An optimal consumption-investment problem has been investigated in a paper by Bardi et al. (2010) where a rigorous asymptotic analysis is performed and where the solution is characterized in the limit of fast volatility dynamics.

In a recent paper, Mariani et al. (2012) proposed a numerical approximation scheme for European option prices in stochastic volatility models including transaction costs based on a finite-difference method. The stochastic volatility dynamics assumed there is a slight generalization of that proposed by Hull & White (1987), since they consider a drift coefficient which is a general (regular) deterministic function of both the time and the underlying asset price, while their diffusion coefficient is linear in the instantaneous volatility.

In the present paper, we propose a different approximation method for European option pricing in stochastic volatility models with transaction costs, based on an asymptotic analysis which follows the approach pioneered by Fouque et al. (2001). The model we consider for the stochastic volatility dynam-ics is of Ornstein–Uhlenbeck type. This model has been originally proposed by Stein & Stein (1991). We provide closed-formulas for the first terms appearing in the asymptotical expansion for European option prices in the limit of fast volatility and small transaction costs. In the present paper, we are not going to provide existence and uniqueness results for the stochastic optimal control problem arising in our pricing methodology, neither we shall discuss rigorously the questions related to asymptotic expansion convergence to the solution required.

The goal of the present investigation is to show how the approach pioneered by Davis et al. (1993) can be extended to stochastic volatility models. To this end we combine with a suitable scaling the asymptotic technique introduced by Whalley & Wilmott (1997) with that proposed by Fouque et al. (2001) in order to obtain quite explicit results. The existence and uniqueness of the solution for the singular control problem arising in American option pricing and in the same modelling framework considered in the present paper has been proved by Cosso & Sgarra (2013). We point out that a recent paper by Bichuch (2011) deals with contingent claim pricing in models with (proportional) transaction costs via an asymptotic analysis based on the pioneer work by Whalley & Wilmott (1997) by providing a rigorous derivation of the asymptotic expansion together with lower and upper bounds for the value function involved. In a previous paper, the same author presented a rigorous asymptotic analysis for

an optimal investment problem, in the case of a power utility function, in finite time for a model with proportional transaction costs (Bichuch, 2012). An approximate hedging strategy construction has been also proposed quite recently in a paper by Lepinette & Quoc (2012) for a local volatility model with proportional transaction costs.

The plan of the paper is the following: in Section 2 we introduce the model considered. In Section 3
we formulate the European option pricing problem as a stochastic control problem, and present the
associated HJB equation. In Section 4, the asymptotic analysis is performed, by assuming small
trans-action costs and fast mean-reverting volatility. In Section 5, the price of the option is computed and in
Section 6, the numerical results and the conclusions are provided. For the reader’s convenience, in
Appendix* A the source term of the equation obtained through the asymptotic analysis at O(ε) is *
cal-culated; in Appendix B the averages with respect to the Ornstein–Uhlenbeck invariant measure using
the stochastic volatility proposed by Chesney & Scott (1989) are presented; finally, in Appendix C the
derivatives, which appears in the obtained corrected pricing formula, with respect to the stock price of
the classical Black and Scholes solution are recalled and collected.

**2. A stochastic volatility model with transaction costs**

We suppose to have the following multi-dimensional stochastic process:

*dB= rB dt − (1 + λ)S dL + (1 − μ)S dM ,* (2.1)

*dy= dL − dM ,* (2.2)

*dS= S(α dt + f (z) dW),* (2.3)

*dz= ξ(m − z) dt + β(ρ dW +*1*− ρ*2_{dZ).}_{(2.4)}
In the above equations B and S are the risk-free (the ‘Bond’) and the risky asset (the ‘Stock’),
respectively,* r is the risk-free interest rate, α is the drift rate of the stock, λ and μ are the *
(propor-tional) cost of buying and selling a stock, f is the volatility function, that we shall suppose to depend on
the* stochastic variable z, which is sometimes called the volatility driving process. L(t) and M (t) are the *
cumulative number of shares bought or sold, respectively, up to time t. Both L(t) and M (t) are
assumed* to be right continuous with left-hand limits, non-negative and non-decreasing Ft-adapted *
pro-cesses,* where Ft*denotes* the filtration generated by (Wt, Zt). Moreover, by convention, we shall assume *
*L(0) = M (0) = 0. We keep the notations introduced in Davis et al. (1993) and Whalley & Wilmott *
(1997), where the reader can find a detailed justification for the transaction costs model just
intro-duced.* The set T of trading strategies π(t), in the present setting, consists of all the two-dimensional, *
right-continuous,* measurable processes (Bπ(t), yπ(t))* which are the solution of (2.2) and for which
some* pair of right-continuous, measurable, Ft-adapted, increasing processes L(t), M (t) exist, such*

that*(Bπ(t), yπ(t), S(t)) ∈ EΓ* *∀t ∈ [0, T], where EΓ= (B, y, S) ∈ R × R × R*+*: B+ c(y, S) > −Γ , with*
*Γ positive constant and c(y, S) the liquidated cash value of the position y, i.e. the residual cash value*
when a long position*(y > 0) is sold or a short position (y < 0) is closed. Conventionally, we assume*
*c(0, S) = 0. In what follows we shall always suppose f (z) to be a function bounded away from 0:*

0*< m*1* f (z) m*2*< ∞, ∀z.*

*The process followed by the stochastic variable z is a Ornstein–Uhlenbeck process with average m.*
The parameter*ξ is the rate of mean-reversion volatility.*

The* Brownian motions W and Z are uncorrelated and ρ is the instantaneous correlation coefficient *
between* the asset price and the volatility shocks. Usually one considers ρ < 0, i.e. the two processes *
are anti-correlated (e.g. when the prices go down the investors tend to be nervous and the volatility
raises). For more details see Fouque et al. (2001) and Jonsson & Sircar (2002).

If we suppose to deal with trading strategies absolutely continuous with respect to time, we can write our stochastic dynamics in the following compact form:

*L*=
*t*
0
*l ds,* *M*=
*t*
0
*m ds.*

Therefore the process we are dealing with can be written in the following form:

*dB= [rB − (1 + λ)Sl + (1 − μ)Sm] dt,* (2.5)

*dy= (l − m) dt,* (2.6)

*dS= S(α dt + f (z) dW),* (2.7)

*dz= ξ(m − z) dt + β(ρ dW +*1*− ρ*2*dZ).* _{(2.8)}
Remark* 2.1 The assumption of absolute continuity with respect to time on the processes L(t), M (t) was *
made in order to write our dynamics in a simpler form, but it can be relaxed. Following the treatment
presented* in Davis et al. (1993), all the results we are going to mention hold true for L(t) and M (t) right *
continuous* with left-hand limits, non-negative, non-decreasing and Ft-adapted.*

If we denote the value at time t of a portfolio of the writer of a European option with strike price K,
after* following the strategy π, by Φw(t, Bπ(t), yπ(t), S(t), z(t)), its final value is given by:*

*Φw(T, Bπ _{(T), y}π_{(T), S(T), z(T)) = B}π_{(T) + I}*

*(S(T)<K)c(yπ(T), S(T))*

*+ I(S(T)>K)[c(yπ(T) − 1, S(T)) + K].* (2.9)
On the other hand, the final value of a portfolio which does not include the option (denoted by*Φ*1)
is simply

*Φ*1*(T, Bπ(T), yπ(T), S(T), z(T)) = Bπ(T) + c(yπ(T), S(T)).* (2.10)

**3. The optimal control problem for European option pricing**

The purpose of this section is to briefly recall the basic ideas of Utility Indifference Pricing and to resume, in a synthetic way, the framework proposed by Davis et al. (1993) for European option pricing in market models with transaction costs. We define the following value functions:

*Vj(B) = sup*

*π∈TE(U(Φj(T, B*

*π _{(T), y}π_{(T), S(T), z(T))))}*

_{(3.1)}

*for j= 1, w. U : R → R is the utility function, which is required to be concave and increasing and *
satis-fying*U(0) = 0. Note how these value functions depend on the initial endowment B.*

Following Davis et al. (1993) we now define:

Bj*= inf{B : Vj(B) 0}.*

*In the present setting the price of the option C, i.e. the amount of money that the writer has to receive*
to accept the obligation implicit in writing the option, will therefore be:

*C*= Bw− B1. (3.2)

For this price the investor would in fact be indifferent between the two possibilities of going into
the market to hedge the option, or of going into the market without the option. We point out that the
*present approach, named Utility Indifference Pricing, provides just a criterion to select one price, among*
the infinitely many compatible with the no-arbitrage requirement, since the present market model is
incomplete.

We can define the following function that will be useful in the sequel

*Ψj(T, Bπ _{(T), y}π_{(T), S(T), z(T)) = Φj(T, B}π_{(T), y}π_{(T), S(T), z(T)) − B}π_{(T).}*

_{(3.3)}

*We have to find an equation for Vj. In what follows we shall suppress the index j and denote Vj*with

*V . The* problem we are dealing with is a stochastic control problem, where the control is the trading
strategy m and l.

By following step by step the procedure illustrated in Davis et al. (1993, pp. 476–477), including the dependence of just one more state variable, it is possible to show that V must satisfy an HJB equation. A rigorous derivation of the (HJB) equation and the existence and uniqueness proof of a solution in the same modelling context, but for the more complex case of American option pricing, can be found in Cosso & Sgarra (2013).

Proposition 3.1 Let the stochastic process be described by (2.5–2.8), then the value function V defined before must satisfy the following (HJB) equation:

max{(∂y*Vj*− (1 + λ)S∂B*Vj), −(∂yVj*− (1 − μ)S∂B*Vj),*
*∂tVj+ rB∂BVj+ αS∂SVj+ ξ(m − z)∂zVj*
+1
2*[f(z)]*
2* _{S}*2

_{∂SSVj}_{+}1 2

*β*2

_{∂zzVj}_{+ βfSρ∂SzVj} = 0.}_{(3.4)}We now consider the case of the exponential utility function

*U(x) = 1 − exp (−γ x). We note, just*

*in passing by, that this gives for Vj*the following expression:

*Vj= 1 − inf{E[exp (−γ B(T)) exp (−γ Ψj)]},*
where*Ψj*has been previously introduced.

*In the above maximization problem let us change the variables passing Vj−→ Qj:*
*Vj*= 1 − exp

−*γ _{δ}(B + Qj)*,
where

Note that with the above expression for Vj, the price of the option C, as given in (3.2), now

becomes C = Q1 − Qw(3.5) and we can state the auxiliary result:

*Proposition 3.2 The maximization problem for Vj*is equivalent to the following minimization problem
*for Qj:*
min
*(∂yQj− (1 + λ)S), (−∂yQj+ (1 − μ)S), ∂tQj− rQj+ αS∂SQj+ ξ(m − z)∂zQj*
+ 1
2*[f(z)]*
2* _{S}*2

_{∂SSQj}_{−}

*γ*

*δ(∂SQj)*2 +1 2

*β*2

_{∂zzQj}_{−}

*γ*

*δ(∂zQj)*2

*+ βfSρ*

*∂SzQj*−

*γ*

*δ∂SQj∂zQj*= 0. (3.6)

As shown in the papers by Davis et al. (1993) and Whalley & Wilmott (1997), the European option
pricing* problem is a free boundary problem and the (S, y, z) space divides into three regions, the Buy *
region, the Sell region and the No Transaction (NT from now on) region. The NT region is separated
from the other two regions by two (unknown) boundaries and the optimal policy of the option writer
consists in maintain his portfolio in the NT region; when an eventual movement of the asset price forces
the portfolio to hit one of the two boundaries, he must trade so as to stay inside the NT region. In the
Buy region the following condition must hold

min(∂yQj*− (1 + λ)S) = 0,* (3.7)

while in the Sell region the following holds:

min(∂yQj*− (1 − μ)S) = 0.* (3.8)

The NT region is characterized by the following equation:
*∂tQj− rQj+ αS∂SQj*+1
*ε(m − z)∂zQj*+
1
2*[f(z)]*
2* _{S}*2

_{∂SS}_{Qj}_{−}

*γ*

*δ(∂SQj)*2 +1

*εν*2

*∂zzQj*−

*γ*

*δ(∂zQj)*2 +√1

*εν*√

*2fSρ*

*∂SzQj*−

*γ*

*δ∂SQj∂zQj*= 0. (3.9)

*Some continuity conditions for the unknown function Q and its first and second derivatives across*
the free boundaries must be imposed in order to solve the free boundary problem. We shall present them
in next section.

**4. Small transaction costs and fast mean-reverting volatility: the asymptotic analysis**

We now suppose small transaction costs and fast mean-reverting volatility. Moreover, we will assume that the transaction costs are much smaller than the rate of mean reversion:

*λ = ¯λε*2_{,} * _{μ = ¯με}*2

_{,}

*1*

_{ξ =}*ε*,

*β =*√ 2ν √

*ε*.

Buying and selling costs are assumed to be the same for simplicity in the following, whenever anything different will be specified.

We believe that our asymptotic assumptions are consistent with a situation where a large investor,
facing very small transaction costs, is involved. In fact, in the empirical study (Fouque et al., 2000), it
is* found that ε ∼ .005. In the literature (Whalley & Wilmott, 1997; Davis et al., 1993; Zakamouline,*
2006a),* typically it is assumed 0.2% λ .01%.*

In* the absence of transaction costs and with a deterministic volatility ε = 0, the investor would *
con-tinuously* trade and get a perfect hedge staying at y = y*∗, the ‘B&S’ hedging strategy. When transaction
costs* are present there is a strip of small thickness around y = y*∗ where he does not transact. To resolve
this strip we introduce the inner scaled coordinate Y

*y= y*∗*+ εaY* and *∂y−→ ε−a∂Y*. (4.1)

The unknown boundaries between the NT region and the buy and sell regions are located at:
*y= y*∗*+ εaY*+ and *y= y*∗*− εaY*−.

It is very important from the practical hedger point of view to determine Y+ _{and Y}−_{. }

We impose the following matching conditions (see e.g. Whalley & Wilmott, 1997)

*Q*NT*(Y = Y*±*) = Q(y = y*∗*± εaY*±*)* continuity,

*∂YQ*NT*(Y = Y*±*) = εa∂yQ(y = y*∗*± εaY*±*)* continuity of the first derivative,
*∂YYQ*NT*(Y = Y*±*) = ε2a∂yyQ(y = y*∗*± εaY*±*) smooth pasting boundary condition.*

*These boundary conditions will force, in the asymptotic analysis below, a*=1_{3}. Therefore the NT
*strip will have a thickness O(ε*1*/3 _{).}*

(4.2)

(4.3)
In* the buy region (Y < Y*−) we have (3.7) and a solution is

*Q * = *(1 * + *λ)Sy * + *H*−*(t, * *S, * *λ).*

In* the sell region (Y > Y*+) we have (3.8), which solves to

*Q = (1 − μ)Sy + H*+*(t, S, λ).*

Here H+* _{(t, S, λ) and H}*−

_{(t, S, λ) are arbitrary functions not depending on y.}In the NT region we have (3.9), whose solution will be obtained and illustrated in the following

sections.

4.1 The solution in the NT region

As we mentioned before, in the NT region we use the rescaled variable Y defined by (4.1). The change

of variable leads to the following transformation rules for the derivatives:
*∂y−→ ε−1/3 _{∂Y}*

_{,}

*∂S−→ ∂S− ε−1/3*∗

_{y}

_{S∂Y}_{,}

*∂t−→ ∂t− ε−1/3*∗

_{y}*t∂Y*,

*∂z−→ ∂z− ε−1/3*∗

_{y}*z∂Y*.

By writing the solution in the NT region in the following form:
*Q*NT*= S(y*∗*+ ε*1*/3Y) + U*0*(S, t, z) +*
13
*i*_{=1}
*εi _{/6}_{Ui(S, t, z) + ε}*14

*/6*14

_{U}*(S, t, z, Y) + · · · ,*(4.4)

*we can easily obtain an expression for all the derivatives of Q*NT*with respect to the variables t, S, z. By*
adopting an obvious notation they can be written as follows:

*∂tQ*NT*= U0t*+
11
*i*_{=1}
*εi _{/6}*

*Uit+ ε*12

*/6(U12t− y*∗

*tU14Y) + · · · ,*

*∂SQ*NT

*= y*∗

*+ U0S+ ε*1

*/6U1S+ ε*2

*/6(Y + U2S) +*11

*i*=3

*εi/6*

*UiS+ ε*12

*/6*

*U12S− y*∗

*SU14Y*+ · · · ,

*∂SSQ*NT

*= U0SS*+ 9

*i*

_{=1}

*εi*

_{/6}*UiSS+ ε*10

*/6(U10S+ (y*∗

*S)*2

*U14YY) + · · · ,*

*∂zQ*NT

*= U0z*+ 11

*i*=1

*εi/6*

*Uiz+ ε*12

*/6(U12z− y*∗

*zU14Y) + · · · ,*

*∂zzQ*NT

*= U0zz*+ 9

*i*=1

*εi*

_{/6}*Uizz+ ε*10

*/6(U10zz+ (y*∗

*z)*2

*U14YY) + · · · ,*

*∂SzQ*NT

*= U0Sz*+ 9

*i*

_{=2}

*εi/6*

*UiSz+ ε*10

*/6(U10Sz+ y*∗

*Sy*∗

*zU14YY) + · · ·*(4.5)

The introduction of the new scaled variables allows to split the description of the Black–Scholes delta-hedging strategy from the effects due to transaction costs and stochastic volatility and to consider separately their contribution.

To calculate the price of the option we shall use (3.5). The price will have the same asymptotic

expansion as Qjwith j = 1, w, namely
*C= C*0*+ ε*1*/6C*1*+ ε*1*/3C*2+

√

*εC*3*+ ε*2*/3C*4*+ ε*5*/6C*5*+ εC*6+ · · · . (4.6)
*Each Ci*is given by:

*Ci= U _{i}*1

*− U*.

_{i}w*To find the appropriate final conditions for the Ci, we write the final conditions for Q*1*and Qw*. They
are, respectively:

*Q*1*(T) = y(T)S(T)* (4.7)

and

Given the expression (4.4) one has that the final conditions for the Uiare the following:

*Ui*1*(T) = 0 for i = 0, . . . , 6.* (4.9)

*U*_{0}*w(T) = − max(S(T) − K, 0),* (4.10)

*Uiw(T) = 0 for i = 1, . . . , 6.*

Remark 4.1 We point out that the final conditions of our original problem have been imposed on the leading order term, while the corresponding conditions imposed for lower-order terms are homoge-neous. This choice seems to be quite natural in the present setting and is also common for asymptotic expansions.

4.2* The O(ε*−1*) up to O(ε−1/6) order equations*

To simplify the notation, and following the use in Fouque et al. (2003) and Jonsson & Sircar (2002),
we* define the linear operators Li*and* the non-linear operator NL*:

*L*0*U= (m − z)Uz+ ν*2*Uzz,* (4.11)
*L*1*U= −ν*
√
2ρ*(α − r)*
*f* *Uz+ ν*
√
*2fSρUSz,* (4.12)
*L*2*U= Ut*+
1
2*f*
2* _{S}*2

_{USS}_{− rU + rSUS,}_{(4.13)}

*NLU= −ν*2

*γ*

*δ(Uz)*2. (4.14)

*The O(ε*−1_{) equation is simply}

*L*0*U*0*+ NLU*0= 0. (4.15)

*The above equation can be considered an ordinary differential equation (ODE) in z for U*0:
*ν*2_{U}

*0zz+ (m − z)U0z− ν*2
*γ*

*δ(U0z)*2= 0. (4.16)

In Jonsson & Sircar (2002), it is proved that the only solution of an equation of this form is a U which does not depend on z. The conclusion we therefore draw is that

*U*0*does not depend on z.*
*Analogously, the O(ε−i/6 _{) equations, for i = 1, . . . , 5 are}*

*L*0*Ui*= 0.
The conclusion is

4.3 *The O(1) equation*
*The O(1) equation writes*

*L*0*U*6*+ L*2*U*0*+ S(α − r)(U0S+ y*∗*S) −*
1
2*[f(z)]*

2* _{S}*2

*γ*

*δ(y*∗*+ U0S)*2= 0. (4.17)
The above equation will be analysed in Section 4.5.

4.4 *The O(ε*1*/6) equation*
*The O(ε*1*/6 _{) equation is}*

*L*0*U*7*+ L*2*U*1*+ S(α − r)U1S− [f (z)]*2*S*2
*γ*

*δU1S(y*∗*+ U0S) = 0.* (4.18)
Also (4.18) will be analysed in Section 4.5.

4.5 *The O(ε*2*/6 _{) equation}*

*The O(ε*2

*/6*

_{) equation is}*L*0

*U*8

*+ U2t− r(SY + U*2

*) + αS(Y + U2S) +*1 2

*[f(z)]*2

*2*

_{S}

_{U}*2SS*− 2

*γ*

*δ(y*∗

*+ U0S)(Y + U2S)*= 0. (4.19)

*In the above equation there are terms that do not depend on Y , and terms linear in Y . They must be*

*equal to zero separately. From the terms linear in Y one gets:*

*y*∗*= −U0S*+

*(α − r)δ*

*f*2*Sγ* . (4.20)

The above expression gives the leading order (in the absence of transaction costs) optimal hedging strategy. One recognizes the Black and Scholes Delta-hedging strategy. The same result appears in the paper by Whalley & Wilmott (1997, formula (3.6)).

By* inserting the above expression into the O(1) (4.17), one gets:*

*L*0*U*6*+ ∂tU*0+
1
2*f*
2* _{S}*2

*0*

_{∂SSU}*+ rS∂SU*0

*− rU*0+ 1 2

*δ*

*γ*

*(α − r)*2

*f*2 = 0. (4.21) (4.22)

The above equation, considered as an ODE for U6, is of the form:

*L*0*U = χ.*

In Fouque et al. (2003) it is shown that the solvability condition for (4.22) is

where the average · is taken with respect to the Ornstein–Uhlenbeck process invariant measure (see Klebaner, 1998 and Fouque et al., 2003, formula (3.3), p. 1652)

*χ
=* 1
*ν*√2π
R*χ(z) e*
*−(m−z)*2* _{/2ν}*2

*dz.*(4.24)

Therefore the solvability condition for (4.21) is

*∂tU*0+
1
2*¯σ*
2* _{S}*2

*0*

_{∂SSU}*− rU*0

*+ rS∂SU*0= −

*δ(α − r)*2 2γ 1

*¯τ*2, (4.25)

where *¯σ is the effective constant volatility*

*¯σ*2* _{= f}*2

_{ ,}and

*¯τ is defined as*1

*¯τ*2= 1

*f*2 .

Once one imposes to (4.25) the appropriate final condition, which will be different for the investor with option liability and the investor without it, then U0 is determined. One can go back to equation

(4.21) and solve it for U6. Once (4.21) has been solved, and the proper boundary conditions are taken

into account, it is easy to recognize that the solution for U6 can be written as a superposition of two

terms,* one independent on z (the differential operator L*2 contains only terms proportional to the first

derivatives with respect to z and this term satisfies the homogeneous equation L2*U* = 0) and one

depend-ing* on (S, z, t), in such a way that we can write*

*U*6*= U*6*(z)(S, z, t) + ˜U*6*(S, t),* (4.26)
*where, U*_{6}*(z)(S, z, t), the part of U*6*which depends on z, has the following expression*

*U*_{6}*(z)(S, z, t) = −L*−1_{0}
1
2*S*
2* _{(f}*2

*2*

_{− ¯σ}

_{)U}*0SS*+ 1 2

*δ*

*γ(α − r)*2 1

*f*2 − 1

*¯τ*2 ;

on the other hand, ˜*U*6*is a function that does not depend on z and that will be determined by the O(ε)*
equation in the asymptotic procedure.

*We can get a more explicit representation for U*_{6}*(z)*, that will be useful in the next subsection. We first
define the functions*ϕ(z) and ψ(z) as the solutions of the following problems:*

*L*0*ϕ = f*2*− f*2
, (4.27)
*L*0*ψ =*
1
*f*2 −
1
*f*2
. (4.28)

*Therefore the above expression for U*_{6}*(z)*can be written as:

*U*_{6}*(z)*= −
1
2*S*
2_{U}*0SSϕ +*
1
2
*δ*
*γ(α − r)*2*ψ*
. (4.29)

(4.30)

Let us come back to (4.18). Once substituted the expression (4.20) for y∗_{, (4.18) reduces to}

*L*0*U*7 + L2*U*1 = 0.

Equation (4.30) is a Poisson problem for U7, whose solvability condition reads:

*L*2*
U*1= 0, (4.31)

*which is a homogeneous Black–Scholes equation for U*1. Given that the final condition is zero, we get
the conclusions

*U*1≡ 0,

*U*7= ˜*U*7*(S, t) does not dependent on z.*

One can now go back to (4.19), collect all terms independent of Y and get the following equation:
*L*0*U*8 + L2*U*2 = 0.

The above equation is a Poisson problem of the type (4.22). The solvability condition is:

*L*2*
U*2= 0. (4.32)

*Note that the above equation is homogeneous in U*2. Given that the final condition, both for the
investor with option liability and for the investor without it, is 0, one gets the following conclusions

*U*2≡ 0,

*U*8*= ˜U*8*(S, t) is independent of z.*

*Therefore, the first significant correction to the Black and Scholes value is the O(ε*1*/2 _{) contribution.}*

4.6* The O(ε*3*/6 _{)}_{ equation}*

Using the expression (4.20) for y∗_{, the}* _{ O(ε}*3

*/6*

_{)}_{ equation can be written as}

*L*2*U*3*+ L*1*U*6*+ L*0*U*9= 0. (4.33)

*˜U*

˜

Note that in the above equation appears U6, which until now we have derived only up to the function
6*(S, t), to be determined by a higher order asymptotic. However, in (4.33) U*6 is* hit by the operator L*1,

which cancels U6*(S, t).*

Therefore, one can consider (4.33) as a Poisson problem for U9, whose solvability condition is

*L*2*
U*3*= −L*1*U*6
. (4.34)

*The above equation is a Black and Scholes equation for U*3with 0 final condition and with a source
term. We now want to rewrite the source term.

By using for U6 the* expression (4.26) the operator L*1 cancels the part not depending on z, and taking

into account the expression (4.29), one can express the source term in (4.34) as

*−L*1*U*6
=
*L*1
1
2*S*
2_{U}*0SSϕ +*
1
2
*δ*
*γ(α − r)*2*ψ*
=√*νρ*
2
*f ϕ** _{
(S}*3

_{U}*0SSS+ 2S*2

*U0SS) − (α − r)S*2

*U0SS*

_{ϕ}_{}

*f*−

*3*

_{γ}δ(α − r)

_{ψ}_{}

*f*.

*Therefore, U*3solves the following Black and Scholes equation

*U3t*+
1
2*¯σ*
2* _{S}*2

_{U}*3SS− rU*3

*+ rSU3S*=√

*νρ*2

*f ϕ*

*3*

_{ (S}

_{U}*0SSS+ 2S*2

*U0SS) − (α − r)S*2

*U0SS*

*ϕ*

*f*−

*3*

_{γ}δ(α − r)*ψ*

*f*(4.35) with zero final data.

4.7 *The O(ε*2*/3 _{) equation}*

*Since U*7*does not depend on z, the O(ε*2*/3) equation can be written as*
*L*2*U*4*+ L*0*U*10*+ ν*2*(y*∗*z)*

2

*U14YY* −
*γ*

*δf*2*S*2*Y*2= 0. (4.36)

Equation (4.36) can be considered as an ODE in Y for U14. It writes as

*U14YY= AY*2*+ B,* (4.37)

where we have defined the following quantities
*A*=*γ*
*δ*
*f*2*S*2
*ν*2* _{(y}*∗

*z)*2 ,

*B*= −

*L*2

*U*4

*+ L*0

*U*10

*ν*2

*(y*∗

*z)*2 . Equation (4.37) solves to

*U*14=

*A*12

*Y*4

_{+}1 2

*BY*2

_{+ CY + D,}_{(4.38)}

*with C and D independent of Y . Now we have to impose the matching conditions.*
Being

*Q*BUY*= (1 + ε*2*¯λ)Sy + H*−*(S, z, t) in the outer buy region*
*Q*SELL*= (1 − ε*2*¯μ)Sy + H*+*(S, z, t) in the outer sell region*

(4.39) and imposing the continuity of the gradient at the two boundaries:

*∂YQ*NT*(Y = −Y*−*) = ε*1*/3∂yQ*BUY*(y = y*∗*− ε*1*/3Y*−*),*
*∂YQ*NT*(Y = Y*+*) = ε*1*/3∂yQ*SELL*(y = y*∗*+ ε*1*/3Y*+*),*

*one gets, at O(ε*14*/6 _{)}*

*∂YU*14*(Y = −Y*−*) = ¯λS,* (4.40)

*∂YU*14*(Y = Y*+*) = − ¯μS.* (4.41)

Therefore, using (4.38), one gets

−*A*
3*(Y*
−* _{)}*3

*−*

_{− BY}

_{+ C = ¯λS,}_{(4.42)}

*A*3

*(Y*+

*3*

_{)}*+*

_{+ BY}

_{+ C = − ¯μS.}_{(4.43)}

*Moreover, being W in the outer regions linear in y, one imposes the continuity of the second *
deriva-tive as follows

*∂YYQ*NT*(Y = ±Y*±*) = 0,*
i.e.

*A(Y*+* _{)}*2

_{+ B = 0,}*A(Y*−

*)*2

*+ B = 0.*

From these equations one sees that, at this order, the bandwidth about the Black and Scholes strategy is
symmetric, i.e.
*Y*+*= Y*−=
−*B*
*A*
1*/2*
. (4.44)

Subtracting the two equations (4.42) and (4.43) to eliminate C, and using the above expressions for Y±_{, }

one gets

4
3*(−B)*

3*/2 _{A}−1/2_{= (¯λ + ¯μ)S.}*

After some manipulations, and using the expressions for A and B, (4.37) leads to the following equation:

*L*0*U*10*+ L*2*U*4=
3
4*(¯λ + ¯μ)fS*
2
*γ*
*δν*2*(y*∗*z)*2
2*/3*
. (4.45)

One can also find an expression for the amplitude of the NT region

*Y*+*= Y*−=
3
2
1
*f*2_{S}*δ*
*γν*2*(y*∗*z)*2
1*/3*
. (4.46)

Equation (4.45) is a Poisson problem of the type of (4.22). The solvability condition gives an

*equation for U*4:
*L*2*
U*4=
3
4*(¯λ + ¯μ)fS*
2
*γ*
*δν*2*(y*∗*z)*2
2*/3*
. (4.47)

Note* also that, adding the two equations (4.42) and (4.43), one gets that C = 0. Therefore*
*U*14=
*A*
12*Y*
4_{+}1
2*BY*
2_{+ D,}_{(4.48)}

which will be useful in Section 4.9.
4.8 The O(ε5*/6 _{)}_{ equation}*

The* O(ε*5*/6 _{)}*

_{ equation writes as:}

*L*0*U*11*+ L*2*U*5−
√
2*∂U*6
*∂z*
*γ*
*δνρf (z)SY −*
*∂U*3
*∂S*
*γ*
*δf(z)*2*S*2*Y* +
*∂*2* _{U}*
15

*∂Y*2

*∂y*∗

*∂z*2

*ν*2

_{= 0.}

_{(4.49)}

*This equation can be considered as an ODE for U*15:
*∂*2_{U}

15

*∂Y*2 *= ¯AY + ¯B,*
where we have denoted

*¯A =*√2*∂U*6
*∂z*
*γ*
*δνρf (z)S −*
*∂U*3
*∂S*
*γ*
*δf(z)*2*S*2
*(ν*2* _{y}*∗2

*z*

*),*

*¯B = −L*0

*U*11

*+ L*2

*U*5

*ν*2

*∗2*

_{y}*z*.

Integrating (4.49) with respect to Y and using the boundary conditions:

*U15Y(Y*+*) = U15Y(−Y*−*) = 0*
which are needed to ensure the continuity of the gradient, one gets

*L*0*U*11*+ L*2*U*5= 0. (4.50)

*The above equation is a Poisson problem for U*11, whose solvability condition reads

*L*2*
U*5= 0. (4.51)

*This is a homogeneous Black–Scholes equation for U*5. Given that the final condition is zero, we get
the conclusions:

*U*5≡ 0,

4.9 *The O(ε) equation*

*Collecting the O(ε) terms one gets*
*L*2*U*6*+ L*1*U*9*+ L*0*U*12−
1
2*f*
2* _{S}*2

*γ*

*δ(U3S)*2

*− ν*√

*2fSργ*

*δU3SU6z− y*∗

*z(m − z)U14Y*−

*γ*2

_{δ}f*YS*2

*U4S+ ν*2

*−y*∗

*zzU14Y* *− 2y*∗*zU14Yz+ (y*∗*z)*2*U16YY*−
*γ*
*δ(U6z)*2

= 0. (4.52)

*The above equation can be considered as an ODE in Y for U*16:
*∂*2_{U}

16

*∂Y*2 *= ˜AY + ˜B,*
where we have defined

*˜A = γ _{δ}*

*f*2

*S*2

*U4S*

*ν*2

*∗*

_{(y}*z)*2 ,

*˜B = −*

*L*0

*U*12

*+ L*1

*U*9

*+ L*2

*U*6− 1 2

*f*2

*2*

_{S}*γ*

*δU3S*2

*− ν*√

*2fSργ*

*δU3SU6z*

*− ν*2

*∗*

_{y}*zzU14Y+ 2y*∗

*zU14Yz*+

*γ*

*δU6z*2

*+ U14Yy*∗

*z(m − z)*

*(ν*2

*∗*

_{(y}*z)*2

_{).}Note that ˜*A does not depend on Y and in ˜B the Y -dependent terms appear only with their derivatives*
*in Y .*

We* integrate (4.52) from −Y*− to Y+.
Let us use the boundary conditions:

*U16Y(Y*+*) = U16Y(−Y*−*) = 0.*

Moreover, being Y+ _{= Y}− _{and from the expression (4.48) it follows that}
*Y*+
*−Y*−*U14YdY*= 0.
By integrating (4.52) we get:
*L*0*U*12*+ L*1*U*9*+ L*2*U*6=
1
2*f*
2* _{S}*2

*γ*

*δ(U3S)*2

*+ ν*√

*2fSργ*

*δU3SU6z+ ν*2

*γ*

*δ(U6z)*2.

*The solvability condition for U*12

*gives the following equation for U*6:

*L*2*
U*6*= −L*1*U*9
+
1
2*¯σ*
2* _{S}*2

*γ*

*δ(U3S)*2

*+ ν*√

*2Sργ*

*δU3SfU6z(z) + ν*2

*γ*

*δ(U6z(z))*2 . (4.53) The main results of this section are the following:

*1. Equation (4.25) for U*0;

*3. Equation (4.32) for U*2 which led us to U2 ≡ 0;

*4. Equation (4.35) for U*3;

*5. Equation (4.47) for U*4;

*6. Equation (4.51) for U*5 which led us to U5 ≡ 0;

*7. Equation (4.53) for U*6;

*8. The expression (4.20) for y*∗_{, the centre of the NT region.}

9. The expression (4.46) for the boundaries of the NT region.
**5. The option pricing**

To calculate the price of the option we now use (3.5), and the asymptotic expansion (4.6) together with the appropriate final conditions for the Ci, which, as we discussed in the previous section, can be obtained by the final conditions for the Ui, (4.9–4.11). Here we can state our main result, which will be proved in detail in the following subsections:

Proposition 5.1 The first seven terms of the asymptotic expansion (4.6) in powers of the adopted
parameter* ε for the European Call option price in the Utility Indifference framework introduced by *
M.A.H. Davis, V.G. Panas and T. Zariphopoulou are provided by formulas (5.3), (5.4), (5.5), (5.8),
(5.9), (5.10) and (5.15) written below.

5.1 The leading order price

*In order to compute the leading order price we have to calculate U*_{0}1 *and U*_{0}*w*where they both satisfy

(4.25). Given the respective final conditions (4.9) and (4.10) one has that

*U*_{0}1*= (T − t)δ(α − r)*
2
2γ
1
*¯τ*2, (5.1)
*U*_{0}*w= (T − t)δ(α − r)*
2
2*γ*
1
*¯τ*2 *− C*
BS_{,} _{(5.2)}

*where C*BS_{is the classical pricing formula for a European call option, i.e.}
*C*BS*(S, t) = SN(d*1*) − K e−r(T−t)N(d*2*),*
where
*d*1=
log(S/K) + (r + (1/2) ¯σ2_{)(T − t)}*¯σ*√*T− t* , *d*2*= d*1*− ¯σ*
√
*T* *− t,*

*and N(z) is the normal cumulative distribution function.*
*From the above expressions for U*_{0}*j*one obtains

5.2 *The O(ε*1*/6 _{) correction}*

*The equation for U*1

1 *and U*1
*w*_{is (}

4.31), a homogeneous Black and Scholes equation. In both cases, the

*final condition is homogeneous. Therefore U*1

1*≡ 0 and U*1*w*≡ 0 and

*C*1*(S, t) = 0.* (5.4)

5.3 *The O(ε*1*/3) correction*

*As for the O(ε*1*/6 _{) terms, the equation for U}*1
2

*and U*

*w*

2 is the homogeneous Black and Scholes equation (4.32), with homogeneous final condition, therefore both U1

2 *and U*2*w*are zero and

*C*2*(S, t) = 0.* (5.5)

5.4 *The O(ε*1*/2) correction*
*The equation for U*1

3 *and U*3

*w*_{is (4.35), in both cases with homogeneous final condition. Using, }
respec-tively, the expressions (5.1) and (5.2) in (4.35), one has that

*U*_{3}1*= (T − t)*√*νρ*
2
*δ*
*γ(α − r)*3
*ψ*
*f*
, (5.6)
*U*_{3}*w= −(T − t)*√*νρ*
2
−*δ*
*γ(α − r)*3
*ψ*
*f*
*− f ϕ** _{
(S}*3

*BS*

_{C}*3S*

*+ 2S*2

*CSS*BS

*)*

*+ (α − r)S*2

*ϕ*

*f*

*CSS*BS . (5.7) Therefore

*C*3

*(S, t) = −(T − t)*√

*νρ*2

*f ϕ*

*3*

_{ (S}*3*

_{∂}*SC*BS

*+ 2S*2

*∂SSC*BS

*) − (α − r)S*2

*∂SSC*BS

*ϕ*

*f*. (5.8)

5.5 *The O(ε*2*/3 _{) correction}*

*The equation for U*1

4 *and U*
*w*

4 is (4.47), in both cases with homogeneous final condition. The source
*term for the two problems is the same: in fact y*∗*z* has the same expression for both problems. Therefore
*U*_{4}*w= U*1

4 and

*C*4*(S, t) = 0.* (5.9)

5.6 *The O(ε*5*/6 _{) correction}*

*The equation for U*1

5 *and U*5*w*is the homogeneous Black and Scholes equation (4.51) with zero final
condition in both cases. Then

*C*5*(S, t) = 0.* (5.10)

5.7* The O(ε) correction*

To* compute the O(ε) correction we have to solve (*4.53). It is a Black and Scholes equation with a source

see (4.26). The same decomposition holds also for C6:
*C*6*= C(z)*6 + ˜*C*6,
where
*C*_{6}*(z)= U*_{6}*(z)1− U*_{6}*(z)w*,
and
*˜C*6*= ˜U*6*(z)1− ˜U*
*w*
6.

We have already computed U6*(z)j*, as given in (4.29), we can therefore calculate C6*(*

*z)*_{. In fact, using }
(4.29) and the expressions (5.1) and (5.2), one gets

*U*_{6}*(z)1*= −1
2
*δ*
*γ(α − r)*2*ψ,* (5.11)
*U*_{6}*(z)w*=1
2*S*
2* _{C}*BS

*SSϕ −*1 2

*δ*

*γ(α − r)*2

*ψ,*(5.12) which gives

*C(z)*

_{6}= −1 2

*S*2

*BS*

_{C}*SSϕ.*˜

We are now left with the task of computing C6.

The equation for C6 can be derived using (4.53). Subtracting the two equations relative to U1
6 and
*U*_{6}*w*one gets
*L*2*
C*6*= −[L*1*U*91*
− L*1*U*9*w*
] +
1
2*S*
2* _{¯σ}*2

*γ*

*δ*

*(U*1

*3S)*2

*− (U*

*w*

*3S)*2

*+ ν*√

*2Sργ*

*δ*

*U*1

_{3S}*fU*

_{6z}(z)1 − U_{3S}wfU_{6z}(z)w*+ ν*2

*γ*

*δ*

*U6z(z)1)*2

*− (U6z(z)w)*2 . (5.13) This equation is a Black and Scholes equation for C6 with source term. In Appendix A, this source term is explicitly computed and (5.13) writes as

*L*2*
C*6*= (T − t)*2*ˆA + (T − t)ˆB + ˆC,* (5.14)
where
*ˆA = −ν*2*ρ*2
4
*γ*
*δS*2*¯σ*2
*f ϕ** _{
(S}*3

*BS*

_{C}*4S*

*+ 5S*2

*C*BS

*SSS+ 4SC*BS

*SS) − (α − r)*

*ϕ*

*f*

*(S*2

*BS*

_{C}*SSS+ 2SC*BS

*SS)*2 ,

*ˆB = −ν*2

*2*

_{ρ}*2*

_{S}*ϕ*

_{f}_{ −}1 2

*(α − r)*

*ϕ*

*f*

*f ϕ*

*3*

_{ (S}*BS*

_{C}*5S*

*+ 8S*2

*C4S*BS

*+ 14SC*BS

*SSS+ 4CSS*BS

*) − (α − r)*

_{ϕ}_{}

*f*

*(S*2

*BS*

_{C}*4S*

*+ 4SCSSS*BS

*+ 2C*BS

*SS)*

−*ν*2*ρ*2
2 *S*
3_{f ϕ}_{
}
*(S*3* _{C}*BS

*6S*

*+ 11S*2

*C5S*BS

*+ 30SC*BS

*4S*

*+ 18CSSS)f ϕ*BS

*−(α − r)(S*2

*BS*

_{C}*5S*

*+ 6SC4S*BS

*+ 6C*BS

*SSS)*

*ϕ*

*f*−

*ν*2

*ρ*2 2

*γ*

*δS*

*f ϕ*

*3*

_{ (S}*BS*

_{C}*4S*

*+ 5S*2

*C*BS

*3S*

*+ 4SC*BS

*SS)(α − r)*

*ϕ*

*f*

*(2SC*BS

*SS*

*+ S*2

*C*BS

*3S)*×

*S*2

*C*BS

*SSϕ*

*f*−

*δ*

*γ(α − r)*2

*ψ*

*f*,

*ˆC = ν*2

*γ*

*δ*−1 4

*S*4

*BS*

_{(C}*SS)*2

*2*

_{ϕ}_{ +}1 2

*δ*

*γ(α − r)*2

*S*2

*CSS*BS

*ϕ*

*ψ*

*− ν*2

*2*

_{ρ}

_{(α − r)}*(α − r)S*2

*BS*

_{C}*SS*

*G*

*f*

*− (S*3

*BS*

_{C}*SSS+ 2S*2

*BS*

_{C}*SS)*

*F*

*f*

*− ν*2

*2*

_{ρ}*4*

_{(S}*BS*

_{C}*4S*

*+ 5S*3

*C4S*BS

*+ 5S*3

*C*BS

*SSS+ 4S*2

*C*BS

*SS)F*

*f − (α − r)(S*3

*CSSS*BS

*+ 2S*2

*CSS*BS

*)G*

*f*. Given the homogeneous final condition, the solution of (5.14) writes as

*C*6=*(T − t)*
3
3 *ˆA + (*

*T− t)*2

2 *ˆB + (T − t) ˆC.* (5.15)

We have used the fact that
*L*2
*(T − t)*3
3 *A*+
*(T − t)*2
2 *B+ (T − t)C*
*= (T − t)*2_{A}_{+ (T − t)B + C +}*(T − t)*3
3 *L*2*A*+
*(T − t)*2
2 *L*2*B+ (T − t)L*2*C*
and the last three terms are zero

*L*2
*Sn∂*
*n _{C}*BS

*∂Sn*

*= Sn*

*∂n*

*∂SnL*2

*C*BS

_{= 0.}

In Appendix C, the reader can find the derivatives of CBS _{with respect to S up to the sixth order. }
In Appendix B, the averages ·
with respect to the Ornstein–Uhlenbeck process invariant measure are
explicitly computed using Scott’s model.

**6. Numerical illustration of the results**

In this section, we present the main results obtained via the asymptotic method. At first, we plot the NT
region for different values of the volatility, ranging from 5 to 60%, both in the case which does not
include the option, denoted by the index 1, and in the case which includes the option, denoted by the
index* w. The volatility is chosen as in the Scott model, f (z) = ez*_{.InFig.} 1, the curves representing the

Black and Scholes strategy y∗ _{in}* _{ the absence of transaction costs and the hedging boundaries, y = y}*∗

_{± }

0 0.2 0.4 0.6 0.8 1 1.2 0 100 200 300 400 500 600 700 800 900 1000 1100 y 0 0.2 0.4 0.6 0.8 1 1.2 0 10 20 30 40 50 60 70 S y S (a) (b) (d) (c) 0 0.2 0.4 0.6 0.8 1 1.2 0 2 4 6 8 10 12 14 16 18 S y 0 0.2 0.4 0.6 0.8 1 1.2 0 1 2 3 4 5 6 7 8 S y

Fig. 1. The NT region in the case which does not include the option. The dotted curve represents the Black and Scholes strategy

*y*∗in the absence of transaction costs, the other two curves represent the hedging boundaries. See the text for the choice of
parameters. Parts (a,b,c,d) refer to different numerical values of the initial volatility.

that these curves are, respectively, given by

*y*∗=*(α − r)δ*
e*2zSγ* , (6.1)
*y*=*(α − r)δ*
e*2zSγ* *± ε*
1*/3*3(¯λ + ¯μ)(α − r)2*ν*2*δ*3
e*6z _{S}*3

*γ*3 1

*/3*. (6.2)

The corresponding curves in the second case are plotted in Fig. 2and their equations are:

*y*∗*= C*BS*S* +

*(α − r)δ*

0 1 2 3 4 5 6 0 20 40 60 80 100 120 140 160 180 200 S y 0 1 2 3 4 5 6 0 2 4 6 8 10 12 S y 0 1 2 3 4 5 6 0 0.5 1 1.5 2 2.5 3 (c) (a) (b) (d) S y 0 1 2 3 4 5 6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 S y

*Fig. 2. The NT region in the case which includes the option. The dotted curve represents the Black and Scholes strategy y*∗in
the absence of transaction costs, the other two curves represent the hedging boundaries. See the text for the choice of parameters.
Parts (a,b,c,d) refer to different numerical values of the initial volatility.

*y= C _{S}*BS+

*(α − r)δ*e

*2zSγ*

*± ε*1

*/3*3(¯λ + ¯μ)(α − r)2

*ν*2

*δ*3 e

*6z*3

_{S}*γ*3 1

*/3*. (6.4)

Both in Figs 1 and 2, the strike price is K = 0.5, the risk-free interest rate is r = 0.07, the drift rate of
the* stock is α = 0.1, the risk aversion is γ = 1, the mean volatility σ¯ = 0.2, the time to expiry is 0.3,*
*¯λ = ¯μ = 1 and ε =* 1

200.

Finally, in Fig.3it is shown the curve representing the classical Black and Scholes price of a
*Euro-pean call option with the first correction obtained at O(ε*1*/2 _{) and the second correction obtained at O(ε).}*

*Here the parameters are chosen as K= 100, r = 0.04, α = 0.1, γ = 1, the time to expiry is 0.25 and*

*ε =*1

200. In Fig.3(a,b) the correlation coefficient is*ρ = 0, therefore the Black and Scholes price and the*
*corrected price at O(ε*1*/2 _{) coincide as follows from the expression (}*

_{5.8}

_{). The solutions are computed}at two levels of the current volatility

*σ*2

*2*

_{= 0.165 and σ}_{= 0.66 and in the range around the money}0.9

*S/K 1.1 the maximum deviation of the asymptotic approximation (at O(ε)) from the price with*

(a)

(c) (d)

(b)

Fig.* 3. Comparison between the classical price for a European call option and the price including the correction up to O(ε*1*/2 _{)}*

and* up to O(ε), at two levels of the current volatility. (a, b) The region around the money, with ρ = 0 and (c, d) the region out of *

the* money, with ρ = −0.3.*

the lower volatility is 2.1% of this price, with the higher volatility is 7.5%. In Fig. 3(c,d) the correlation
coefficient* is ρ = −0.3. We note that in the region out of the money the second correction of the price *
obtained* at O(ε) improves the approximation obtained at O(ε*1*/2 _{). We want to remark that the }*

oscilla-tory behaviour exhibited in Fig. 2(b–d) by y for small values of S has been already observed by
Whalley & Wilmott (1997) also in models with constant volatility, although this feature seems to be
more pro-nounced in the present context. Moreover, the thickness of the NT region in the presence of
stochastic* volatility seems to be bigger than in model with constant volatility. The (ε)*1*/3 *scaling was

also a relevant feature already established in Whalley & Wilmott (1997) which is exhibited also by the present model.

**Acknowledgements**

**Funding**

The work by GG was partially supported by GNFM/INdAM through Progetto Giovani Grant.

References

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**Appendix**** A. The source term in the equation for C****6**

We separately compute the four terms appearing into the source of (5.13) for C6.
A.1 *The termL*1*U*91*
− L*1*U*9*w*

In the source term we have [L1*U*91*
− L*1*U*9*w
]. We now write L*1*U*
*j*

9*
for j = 1, w, and for notational*
*simplicity we omit the index j.*

Consider (4.33) and solve it for U9:

*U*9*= −L*−10 *(L*2*U*3*+ L*1*U*6*).* (A.1)
One can go back to (4.35) for the term*L*2*U*3:

*L*2*U*3=
1
2*S*
2_{U}*3SSL*0*ϕ +*
*νρ*
√
2
*f ϕ** _{
(S}*3

_{U}*0SSS+ 2S*2

*U0SS)*

*− (α − r)S*2

_{U}*0SS*

*ϕ*

*f*−

*3*

_{γ}δ(α − r)*ψ*

*f*.

Being* L*1*U*6 = L1*U*6*(z)*, from (4.26) it follows that

*L*1*U*6*(z)*=
*νρ*
√
2
*−f ϕ** _{(2S}*2

_{U}*0SS+ S*3

*U0SSS) + (α − r)S*2

*U0SSϕ*

*f*+

*δ*

*γ(α − r)*3

*ψ*

*f*.

*Defining F, G and H as the solutions of the following Poisson equations:*
*L*0*F(z) = f ϕ**− f ϕ*
,
*L*0*G(z) =ϕ*
*f* −
*ϕ*
*f*
,
*L*0*H(z) =ψ*
*f* −
*ψ*
*f*
,

the formula (A.1) writes as

*U*9= −
1
2*S*
2_{U}*3SSϕ +*
*νρ*
√
2
*(S*3_{U}*0SSS+ 2S*2*U0SS)F(z) − (α − r)S*2*U0SSG(z) −*
*δ*
*γ(α − r)*3*H(z)*
.

The first term in the right-hand side of (5.13) is
*L*1*U*91*
− L*1*U*9*w
= ν*2*ρ*2*S*2*(T − t)*
*ϕ*_{f}_{
−}1
2*(α − r)*
_{ϕ}_{}
*f*
*f ϕ** _{
(S}*3

*BS*

_{C}*5S*

*+ 8S*2

*C*BS

*4S*

*+ 14SC*BS

*SSS+ 4C*BS

*SS) − (α − r)*

_{ϕ}_{}

*f*

*(S*2

*BS*

_{C}*4S*

*+ 4SCSSS*BS

*+ 2C*BS

*SS)*

*+ ν*2

*2*

_{ρ}

_{(α − r)}*(α − r)S*2

*BS*

_{C}*SS*

*G*

*f*

*− (S*3

*BS*

_{C}*SSS+ 2S*2

*BS*

_{C}*SS)*

*F*

*f*+

*ν*2

*ρ*2 2

*S*3

_{(T − t)f ϕ}_{ }

*3*

_{(S}*BS*

_{C}*6S*

*+ 11S*2

*BS*

_{C}*5S*

*+ 30SC*BS

*4S*

*+ 18C*BS

*SSS)f ϕ*

_{ }

*− (α − r)(S*2

*BS*

_{C}*5S*

*+ 6SC5S*BS

*+ 6SC*BS

*4S*

*+ 6CSSS)*BS

*ϕ*

*f*

*+ ν*2

*2*

_{ρ}*4*

_{(S}*BS*

_{C}*4S*

*+ 5S*3

*C4S*BS

*+ 5S*3

*CSSS*BS

*+ 4S*2

*C*BS

*SS)F*

*f*

*− (α − r)(S*3

*BS*

_{C}*SSS+ 2S*2

*C*BS

*SS)G*

*f*. A.2

*The term(U*1

*3S)*2*− (U*
*w*
*3S)*2

Using the expressions (5.6) and (5.7), respectively, for U1

3 *and U*3*w*, it follows that
*(U*1
*3S)*
2_{− (U}w*3S)*
2_{= −}*ν*2*ρ*2
2 *(T − t)*
2
*f ϕ** _{
(S}*3

*BS*

_{C}*4S*

*+ 5S*2

*BS*

_{C}*SSS+ 4SC*BS

*SS)*

*− (α − r)*

*ϕ*

*f*

*(S*2

*BS*

_{C}*SSS+ 2SCSS*BS

*)*2 .

A.3 *The term U _{3S}*1

*fU*

_{6z}(z)1 − U_{3S}wfU_{6z}(z)w*Since U*

_{3}1

*is independent of S:*

*U _{3S}*1

*fU*= 0. Using the expression (4.29), we have

_{6z}(z)1*U _{3S}wfU_{6z}(z)w*
=

*νρ*2√2

*(T − t)*

*f ϕ*

*3*

_{ (S}*BS*

_{C}*4S*

*+ 5S*2

*C3S*BS

*+ 4SC*BS

*SS) − (α − r)*

*ϕ*

*f*

*(2SC*BS

*SS*

*+ S*2

*C*BS

*3S)*×

*S*2

*CSS*BS

*ϕ*

*f*−

*δ*

*γ(α − r)*2

*ψ*

*f*.

A.4* The term (U*_{6}*(z*
*z)1 _{)}*

_{2 }

− (U6*(z*
*z)w _{)}*

_{2}

_{ }

Using once again the formula (4.29) it follows

*(U6z(z)1)*2*− (U6z(z)w)*2
= −
1
4*S*
4* _{(C}*BS

*SS)*2

*ϕ*2 + 1 2

*δ*

*γ(α − r)*2

*S*2

*C*BS

*SSϕ*

*ψ*.

**Appendix B. Calculation of the averages·
for Scott’s model**
*If one uses, for the volatility f(z) the model of Scott, i.e.*

*f(z) = ez*,

then one can compute explicitly

*f*2_{
= e}*m+ν*2
,
1
*f*2
= e*m−ν*2
,
_{ϕ}_{}
*f*
=* _{ν}*1

_{2}

*(em*2 − e

_{+(1/2)ν}*m*2

_{+(5/2)ν}*),*

*ψ*

*f*=

*1*

_{ν}_{2}

*(e(3/2)(−2m+3ν*2

*− e*

_{)}*m*2

_{−(3/2)ν}*),*

*ψ*

_{f}_{ =}1

*ν*2

*(e*3

*(m−(1/2)ν*2

*− e*

_{)}*(1/2)(ν*2

_{−2m)}*),*

*f ϕ*

_{ =}1

*ν*2

*(e*

*3m+(5/2)ν*2 − e

*3m+(9/2)ν*2

*),*

*f = em+(1/2)ν*2 , 1

*f*= e

*−m+(1/2)ν*2 ,

*f*2

_{ϕ}_{ =}1 2ν2

*(e*4

*(m+ν*2

*− e4*

_{)}*(m+2ν*2

_{)}*),*

*ϕ*

*f*2 = 1 2

*ν*2

*(1 − e*4

*ν*2

*),*

*ϕ*

_{ = −2 e}2

*(m+ν*2

*,*

_{)}*fF*

_{ =}1

*ν*4 e

*4m+3ν*2− e

*4m+5ν*2+1 2e 4

*(m+2ν*2

*−1 2e 4*

_{)}*(m+ν*2

*,*

_{)}*fG*

_{ =}1

*ν*4

*(2ν*2

_{e}2

*(m+ν*2

*+ e*

_{)}*2m+ν*2 − e

*2m+3ν*2

*),*

*F*

*f*= 1

*ν*4

*(−2ν*2

_{e}2

*(m+ν*2

*+ e*

_{)}*2m+5ν*2 − e

*2m+3ν*2

*),*

*G*

*f*= 1

*ν*4 1 2− 1 2e 4

*ν*2 + e3

*ν*2 − e

*ν*2 ,

*H*

*f*= 1

*ν*4 −1 2 + 1 2e 4

*(2ν*2

*− e*

_{−m)}*−4m+5ν*2 + e

*−ν*2 .

Finally, the term*ϕ**ψ*
can be written as
*ϕ*_{ψ}_{
=} 2 e4*ν*
2
√_{πν}_{2}
_{+∞}
−∞ e
*u*2

*(erf(u −*√2*ν) − erf(u))(erf(u +*√2*ν) − erf(u)) du.*

The value of this integral is numerically calculated. In particular, for*ν = 1/*√2 we obtain*ϕ**ψ*
=
−24.8229.

Analogously, we rewrite*ϕ*2
in the following form
*ϕ*2_{
=}2 e4*(m+ν*
2* _{)}*
√

_{πν}_{2}

_{+∞}−∞ e

*u*2

*(erf(u −*√2

*ν) − erf(u))*2

*du*

and we numerically compute it. For*ν = 1/*√2, it is*ϕ*2
= 17.329 e2*(2m+1)*_{.}
**Appendix C. The derivatives of C****BS**

*The derivatives of C*BS_{with respect to S, up to the sixth order, are explicitly calculated as follows}*∂SC*BS*= N(d*1*),*
*∂SSC*BS= e
*−d*2
1*/2*
*S¯σ*√2π(T − t),
*∂*3
*SC*
BS_{=} −e*−d*
2
1*/2*
*S*2* _{¯σ}*√

_{2}

*1+*

_{π(T − t)}*d*1

*¯σ*√

*T− t*,

*∂*4

*SC*BS= e

*−d*12

*/2*

*S*3

*√*

_{¯σ}_{2}

_{π(T − t)}*d*

_{1}2− 1

*¯σ*2

*(T − t)*+

*3d*1

*¯σ*√

*T− t*+ 2 ,

*∂*5

*SC*BS= e

*−d*12

*/2*

*S*4

*√*

_{¯σ}_{2}

*−*

_{π(T − t)}*d*1

*¯σ*√

*T− t*+ 3

*d*

_{1}2− 1

*¯σ*2

*(T − t)*+

*3d*1

*¯σ*√

*T− t*+ 2 + 1

*¯σ*2

_{(T − t)}*2d*1

*¯σ*√

*T− t*+ 3 ,

*∂*6

*SC*BS= e

*−d*12

*/2*

*S*5

*¯σ*√2π(T − t)

*d*2 1 − 1

*¯σ*2

*+*

_{(T − t)}*3d*1

*¯σ*√

*T− t*+ 2

*d*1

*¯σ*√

*T− t*+ 3 ×

*d*1

*¯σ*√

*T− t*+ 4 − 1

*¯σ(T − t)*+ 2

*¯σ*4

*2 −*

_{(T − t)}

_{¯σ}_{2}

*1*

_{(T − t)}*2d*1

*¯σ*√

*T− t*+ 3

*2d*1

*¯σ*√

*T− t*+ 7 .